Given a set u of 3 vectors in P[sub:1nmgfusa]2[/sub:1nmgfusa](R) p[sub:1nmgfusa]1[/sub:1nmgfusa],p[sub:1nmgfusa]2[/sub:1nmgfusa],p[sub:1nmgfusa]3[/sub:1nmgfusa] where p[sub:1nmgfusa]1[/sub:1nmgfusa](2)=p[sub:1nmgfusa]2[/sub:1nmgfusa](2)=p[sub:1nmgfusa]3[/sub:1nmgfusa](2)=0. Can u be linearly independent? Can it span P[sub:1nmgfusa]2[/sub:1nmgfusa](R)?
My thought is no, it can't be. I tried a few possibilities to see if I could get a set that spanned the typical basis of P[sub:1nmgfusa]2[/sub:1nmgfusa](R) (1,x,x[sup:1nmgfusa]2[/sup:1nmgfusa]) and couldn't get anything. It looks like the fact that these three vectors are equal at x=2 is enough to keep them from being linearly independent and spanning P[sub:1nmgfusa]2[/sub:1nmgfusa](R). I'm not sure how to argue this. Is there a geometric interpretation for linear independence of a set of functions that can help me show that the given condition is enough to make the set linearly dependent and unable to span the vector space. I recognize that if I prove one or the other, then I can prove the second one easily by appealing to the fact that dimP[sub:1nmgfusa]2[/sub:1nmgfusa](R)=3
My thought is no, it can't be. I tried a few possibilities to see if I could get a set that spanned the typical basis of P[sub:1nmgfusa]2[/sub:1nmgfusa](R) (1,x,x[sup:1nmgfusa]2[/sup:1nmgfusa]) and couldn't get anything. It looks like the fact that these three vectors are equal at x=2 is enough to keep them from being linearly independent and spanning P[sub:1nmgfusa]2[/sub:1nmgfusa](R). I'm not sure how to argue this. Is there a geometric interpretation for linear independence of a set of functions that can help me show that the given condition is enough to make the set linearly dependent and unable to span the vector space. I recognize that if I prove one or the other, then I can prove the second one easily by appealing to the fact that dimP[sub:1nmgfusa]2[/sub:1nmgfusa](R)=3